Analysis Seminar: "On the Generalizations of Forelli's Theorem"

{\bf Theorem.} If the function F is defined in the unit ball B in complex Euclidean space of complex dimension n satisfies the conditions:

(1) it is smooth at the origin, and

(2) it is holomorphic when restricted along any complex linear disc passing through the origin then, F is holomorphic on B.

This is perhaps second only to The Hartogs Analyticity Theorem, as far as the complex analyticity criteria are concerned. After a long silence, this theorem has been studied and generalized in several directions. I would like to report some of the recent developments. [cf. Chirka, Proc. Steklov Inst. Math. (2005/2006); Kim-Poletsky-Schmalz, J. Geom. Anal (2009); Joo-Kim-Schmalz, Math. Ann. (2013 & 2016).]